higher-order corrections to the relativistic perihelion advance and the mass of binary pulsars

8/9/2019 Higher-order corrections to the relativistic perihelion advance and the mass of binary pulsars.

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Higher-order corrections to the relativistic perihelion advanceand the mass of binary pulsars.

Maurizio M. DEliseo

Osservatorio S.ElmoVia A.Caccavello 22, 80129 Napoli Italy

We study the general relativistic orbital equation and using a straightforward perturbation methodand a mathematical device rst introduced by dAlembert, we work out approximate expressionsof a bound planetary orbit in the form of trigonometrical polynomials and the rst three terms of the power series development of the perihelion advance. The results are applied to a more precisedetermination of the total mass of the double pulsar J0737-3039.

I. INTRODUCTION

The general relativistic orbital equation for a planet re-volving around a star is deduced from the Schwarzschildline element

ds 2 = c2 dt 2 1 dr 2 r

2 d2 ,

= 1 2r

/ r , d2

= d2

sin2

d2

, (1)where r = /c 2 and = GM are the gravitationalradius and the standard gravitational parameter of thestar, respectively. All remaining symbols have their usualmeaning for this type of problem. According to thegeodesic hypothesis, the path followed by the planet, con-sidered as a test-body not to disturb the metric, can bedetermined using the time-like Lagrangian

2L =dsd

2

= c2 t 2 1

r 2

r 2 (2 + sin 2 2 ), (2)

and a variational principle that uses the functional

S [q] = 2

1L(q, q)d, q = dq/d, (3)

where is the planets proper time and the functionq = q( ) collectively denotes the degrees of freedom of a possible generic planetary motion. The path actuallyfollowed is what makes S [q( )] stationary, i.e. the func-tional derivative

S [q( )]q( )

= 2

1

L[q( ), q( )]q( )

d = 0 , (4)

is zero when computed on the effective motion. From the

working point of view it is well known that a cancelation,in the integrand, of the functional derivative is equivalentto that of the Euler-Lagrange derivative

q

= q

dd

q

, (5)

and so we obtain four second-order differential equations(the Euler-Lagrange equations) determining the soughttime-like geodesic

Lq

dd

L q

= 0 , q = r ,, , t , q = dq/d. (6)

It is worth noting that from the dening equation of theproper time, ds/d = c, we have 2L = c2 , namely theLagrangian is a constant of the motion.

We observe that the Schwarzschild metric, and the La-grangian, are invariant under the reection

(t, r , ,) (t, r , ,), (7)at the hyperplane = / 2, so the mirror image of ageodesic curve clearly has the same property. In par-ticular, if we consider a geodesic which at = 0 startswithin the symmetry hyperplane and is tangent to it,it must coincide with the transformed geodesic, sincethe initial values of position and velocity determine ageodesic uniquely. These considerations are conrmedby the analysis of the Euler-Lagrange equation for

+2r

r

2

2sin2 = 0 , (8)

which admits the solution = / 2 satisfying the initialconditions 0 = / 2, 0 = 0. If we reorient the coordinatesystem so that these conditions are met, the motion takesplace in the equatorial plane, and we can simplify theLagrangian (2) assuming sin = 1, = 0

2L = c2 t 2 1

r 2

r 2

2 . (9)

The coordinates and t are both cyclical: they appearin the Lagrangian only in dotted form, and this meanstwo conservation laws. For the azimuth we have

d

d L

= 0

L

= r 2 = h = const ., (10)

while for t we have

dd

L t

= 0

L t

= c2 t = = const . . (11)

The two constants are related to the angular momentumand to the energy, respectively. Insertion of the two in-tegrals into Eq. (9) leads to

2L = 12

c2 1

r 2

h2r 2

. (12)


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In place of the Euler-Lagrange equation for r , it is easierto derive the radial equation from the integral 2 L = c2 ,obtaining so after multiplication by

r 2 +

h2r 2

2r

2h2 r

r 3+ c2

2

c2= 0 . (13)

We need to cast the path equation into a form which

clearly displays the fact that we are dealing with a Ke-plerian orbit subjected to small relativistic corrections,so we nd it convenient to eliminate the variable andintroduce the angle instead. This is possible since does not enter the equation directly, but only via its dif-ferential d , so we transform the -derivative into a -derivative, according to the identity d/d = ( h/ r 2 )d/d .Moreover, the formulas simplify considerably if we re-place r by its reciprocal u = 1 / r . Denoting by a primedifferentiation with respect to , we have r 2 = h2 u 2 andEq. (13) becomes

u 2 + u2

2

h2 u

2r u3 +

1

h2 c

2

2

c2 = 0 , (14)

while with a differentiation with respect to we nd

u u + u h2 3r

u2 = 0 . (15)

A solution of this equation is u = 0, or u = const., andthis means a circular orbit. Ruling out this possibility,Eq. (15) requires that

u + u =h2

+ 3 r u2 . (16)

Comparing this equation with the classical Binets for-

mula for a particle subjected to a central force of magni-tude f (u) in polar coordinates

u + u = f (u)h2 u2

, (17)

it appears that a particle moving in the Schwarzschildeld will behave as though it were under the inuenceof an effective Newtonian inverse-square force plus anadditional fourth-power inverse force

r 2+

3r h2r 4

r

r , (18)

in a framework in which proper time is used as indepen-dent variable. We remark that for planetary trajectories,in Eq. (16) the terms u and /h 2 are comparable, while uand 3 r u2 differ by a factor 3 r u. The maximum value of this quantity corresponds to the planet which is nearestto the star, so, for Mercury, with r 1.476 105 cm, r 5.51012 cm, it is 3 r

/ r 10 7 . This implies that Eq. (16)

represents in general an oscillator with a constant forcingterm and a weak quadratic nonlinearity which affects itsfrequency 16 .

The orbital equation (16) is not the complete solutionto motion problem, which would require the knowledge

of the function (t), being t the coordinate time, whichis the time measured by an observer at rest at great dis-tance from the origin, and therefore the quadrature of the equations

d d

=1

hu 2,

dtd

=

hu 2, (19)

but this does not concern us here.The importance of Eq. (16) is due to a variety of rea-sons. It originates, without any approximation, from anexact solution of Einsteins equation. Because of the lack-ing of a time variable, it is entirely geometric, so it canbe employed to deduce the precise shape of the orbit andone of the most important post-Newtonian predictions of the theory: the advance of the perihelion of an ellipticalorbit 7 . The equation, then, is naturally linked to Newto-nian dynamics, that in the description of a planetary mo-tion provides already an excellent approximation as wellas a clear connection with observation. Last, it is simplewhen compared with other post-Newtonian equations of

planetary dynamics, and therefore its mathematical andtheoretical limits can be clearly dened. We will use it tocalculate the orbit of a test particle in the gravitationaleld external to a non-spinning spherical mass eventu-ally to the order 1 /c n for any arbitrary positive integern , and the corresponding formulas of the periastron ad-vance to the same order. If the effects predicted fall inthe range of the observability, measurability or indirectdetermination for those physical systems where the equa-tion is applicable, then no doubt they should necessarilybe taken into account in all specic cases.

A determination of higher-order terms of the perias-tron advance of a binary pulsar by using the second

post-Newtonian (2PN) method have been effected bysome auctors 5,18 . Their results based on the 2PN theorycan be applied to the problem of a test particle undera strong gravitational eld by letting the mass of onepulsar theoretically approach zero. Since one can argueabout the rigor of this and other methods used to han-dle the 2PN problem of motion, we think that a properanalysis of the correctness and of the limits of accuracyof these approaches should be based on the agreementunder some convenient limit with the results exposedin the present paper. When considering relativistic ef-fects, other post-Keplerian phenomena come into playin the motion of bodies. For example, one could insert

the relativistic force (18) as a perturbing radial acceler-ation in Gauss equations for the variations of the Kep-lerian orbital elements 1 . In particular, in the expressionof the time derivative of the mean anomaly, in additionto the radial force it appears explicitly the motion of the perihelion. We could cite at this regard two recentworks 10,11 on the secular advance of the mean anomaly inbinary systems, which could be easily extended to includethe higher-order effects calculated in this paper. It wouldalso be interesting to compare the results so obtainedwith those computed by means of the post-NewtonianLagrangian planetary equations 3 , and this could be the


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subject of future work.

II. THE ITERATION METHOD

In the following we will express h in terms of elliptic el-ements of Newtonian approximation, so h = p, where p = a(1

e2 ) is the semi-latus rectum, a is the semi-major

axis and 0 e < 1 is the eccentricity 6 . Although we haveexcluded before the circular orbit, the orbital equationencompasses also this possibility. To make its structuremore apparent and to facilitate the calculations, we castEq. (16) in dimensionless form, which represents a well-known problem in mathematical physics

u + u = 1 + u2 , (20)

where this time u means p/ r and is the pure number3r /p . Thus, when r = a(1 e), u = 1 e. To ndan approximate solution to this equation, one could useregular perturbation theory, writing u in the form of aperturbative expansion in powers of , but it runs in trou-ble here, since we wish an expansion that converges forall values of the independent variable . To do this inthe most convenient way, we perform a shift in the originw = u 1, in order to write the equation as that of theperturbed harmonic oscillator

w + w = (1 + w)2 . (21)

When = 0 the equation becomes

w0 + w0 = 0 , (22)

whose general solution isw0 () = w0 (0)cos + w0 (0) sin . (23)

We shall adopt the initial conditions w0 (0) = e, w0 (0) =0, which hereafter we shall denote standard, and so

w0 () = e cos. (24)

This choice of the initial conditions leads to a simplica-tion of the algebra. Solution (24) has frequency one andperiod 2. The equation of the orbit is then

w0 + 1

p=

1r

=1 + e cos

p, (25)

that is an ellipse, where the angle = 0 locates theposition of the perihelion (because there the function w0is maximal), and identies the direction of the apse line,the greatest symmetry axis of the orbit in the plane.

If we try a straightforward iterative perturbationscheme to solve Eq. (21) attaching the appropriate sub-scripts (the iteration numbers) to its sides, for the rst-order solution w1 we obtain the equation

w1 + w1 = (1 + w0 )2 , (26)

of a forced pendulum where there is a resonant cos term. If we do not want that the pendulum oscillateswith an ever increasing period ( w1 must stay small forall values of ), then the external force is not allowed tohave a Fourier component with the same periodicity asthe pendulum itself. Note here and in the following that,according to the method of undetermined coefficients, theparticular solution of an equation of the form

w + k2 w =n

X n cos nk, (27)

is formally expressed by

w =X 0k2

+X n cos nkk2 (1 n2 ) n =1

+n> 1

X n cos nkk2 (1 n 2 )

, (28)

whose second term is singular, but can be regularizedsince, applying the LHospitals rule, we have

limn 1

X n cos nk

k2

(1 n2

)=

X 1 sin k

2k, (29)

with explicitly present outside the argument of thetrigonometrical function, and therefore growing withoutlimits with time t, since is a monotonic function of t . Obviously this would destroy the stability of the or-bit. Therefore this simple perturbation scheme does notwork for Eq. (21). Things go differently if we rearrangeEq. (21) in the form

w + k2 w = (1 + w2 ), k2 1 2 . (30)The equation is unchanged, but now we have chosen tolook at the isolated linear term on the right side as part

of the unperturbed equation, which isw0 + k

2 w0 = 0 . (31)

Equation (31) now represents an oscillator whose naturalfrequency is k , smaller than 1, and its solution obeyingto the standard initial conditions is

w0 () = e cos k . (32)

It follows that, with respect to the Newtonian oscilla-tor (22), in the relativistic case, to the lowest order, thevalues of r , which trace out an approximated ellipse, donot begin to repeat until somewhat after the radius vec-tor has made a complete revolution. Hence the orbit maybe regarded as being an ellipse which is slowly rotating.In particular, the angular advance for revolution of theapse line is given by

= 2 1k 1 = 2 + O(

2 ). (33)

This is Einsteins perihelion formula, but it representsonly the rst term of a series development in powers of . Our aim is to compute this series up to order 3 . Al-

though the rst approximation gives a satisfactory degree


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of accuracy for ordinary planetary problems, and the sec-ond one can cope with particular astrophysical situations,we push a step further the computation because with lit-tle extra work we shall exhaust all conceivable theoreticalneeds before the orbital equation fails to represent therelativistic motion of bodies in strong-eld situations.

Let us try again the perturbation scheme on Eq. (30).We have now

w1 + k2 w1 = (1 + w20 ), (34)

= 1 +12

e2 +12

e2 cos2k , (35)

that we solve with the standard initial conditions. It isto be noticed that this time we have not any more theresonant term cos k . This equation is of the type

w1 + k2 w1 = A + B cos2k t. (36)

The solution is the sum of the general solution of the ho-mogeneous part and of a particular integral of the com-plete equation. This integral can be found using for-mula (28). Thus the solution of Eq. (36) is

w1 = E cos k + Ak 2 Bk 2

3cos2k . (37)

As w1 is of order as A, B are, we put k 2 = 1, whilethe constant E is determined by the standard initial con-ditions to be

E = e 1 +e2

3, (38)

and so we nd

w1 = 1 +

1

2e2

+ e 1 +1

3e2

cos k

16

e2 cos2k , (39)

which we shall write in abridged form

w1 = A1 + ( e + E 1 )cos k + B 1 cos2k , (40)

where as a notational aid we agree, here and in the fol-lowing, that the subscript number i attached to a capitalletter representing a coefficient wants to emphasize thepresence of the factor i . This solution is periodical andbounded for all values of , and represents the generalrelativistic rst-order deviation from the classical ellip-tical orbit. In the next iteration, which should give thesolution correct to order 2

w2 + k2 w2 = (1 + w21 ), (41)

the right-hand side will present again a resonant term,since to this order we get

w2 + k2 w2 = A + H 2 cos k + B cos2k + C cos3k ,

(42)

where H 2 e(2A1 + B1 ) =2 e(12 + 5 e2 )

6, (43)

with A,B,C to be written later. To get rid of the reso-nant term H 2 cos k on the right side of Eq. (42) we willuse a device which stems naturally from the logical pathwe followed in writing Eq. (30) and that was introducedfor the rst time by dAlembert to control the plague of the secular terms present in the integration of the equa-tion of the lunar motion, which is of the same type of that we are considering here 4 . In order to suppress theunwanted cosine term, we add a counter term and writeEq. (30) in the form

w + k2 H 2e

w = (1 + w2 ) H 2e

w, (44)

and consequently we consider the approximate equation

w2 + k2

2 w2 = (1 + w21 ) H 2e

w1 , (45)

k22 k2

H 2e

. (46)

In the right side, the added term suppresses the resonantterm, since replacing w1 with e cos k 2 (the other termsof w1 would give terms of order greater than 2 ), we ob-tain just H 2 cos k 2 . In the left side the coefficient k2will be diminished by the amount H 2 /e , so that we get

k22 = 1 2 12 + 5e2

62 . (47)

This way we shall obtain an acceptable solution w2 of Eq. (41) to order 2 and, at the same time, the correc-tion to the frequency to the same order. The procedurecan obviously be repeated until we have reached the re-quired approximation degree for both solution and fre-quency. The essence of this method was rediscoveredby Lindstedt 14 and its practical application was furtherelaborated by Poincare 16 . Equation (45) has the form

w2 + k22 w2 = A + B cos2k 2 + C cos3k 2 , (48)

A (2 + e2 )

2 (3e + e3 )

32 , (49)

B e2

2 3e + e3

32 , (50)

C e3

62 , (51)

and its general solution, by Eq. (28), is

w2 = E cos k 2 + Ak 22 Bk 22

3cos2k 2

Ck 22

8cos3k 2 , (52)

where now it suffices to put k 22 = 1 + 2 . We thusnd, by determining E by means of the standard initialconditions, the second-order approximation to the orbit

w2 = A1 + A2 + ( e + E 1 + E 2 )cos k 2 + ( B1 + B2 )cos2k 2 + C 2 cos3k 2 , (53)


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where

A2 = 26 3e + 3 e2 e3

3,

E 2 = 229e3 96e2 + 96 e 288

144,

B2 = 2e3 3e2 + 3 e

9,

C 2 = 2e3

48.

Let us consider now the next equation

w3 + k22 w3 = (1 + w22 )

H 2e

w2 . (54)

Since we are interested only in the resonant term of O( 3 ) we do not solve this equation, but we can extractquickly from the right side the secular generating termH 3 cos k 2 (see Appendix), where

H 3 = e(2A2 + B2 ) = 3 e(36 15e + 15 e2 5e3 )9 . (55)Following dAlemberts method, this term will be can-celed writing Eq. (54) in the form

w3 + k23 w3 = (1 + w22 )

H 2 + H 3e

w2 , k23 = k22 H 3e

,(56)

and we have

k23 = 1

2

12 + 5e2

6

2

36 15e + 15 e2 5e3

9

3 .(57)

Further, we nd

k 3 = 1 18 + 5e2

122

126 30e + 45 e2 10e3

363 ,

(58)

k 13 = 1 + +30 + 5e2

122 +

270 30e + 75 e2 10e336

3 .(59)

The rotation for revolution of the periapse, by denotingwith its angular measure, is given, to order 3 , by

= 2 k 13 1 = 2 + 5 1+16

e2 2

+ 5 3 13

e +56

e2 19

e3 3 . (60)

This formula agrees with the results obtained with thePoincare-Lindstedt method and some of its equivalentmodications 8,15 . By denoting with P the anomalisticperiod, that is the time that elapses between two pas-sages of the object at its perihelion expressed in days,

the average advance rate is

(rad/d) = + P 1 + 2 + 3

= +6 r

a(1 e2 )P +

15 r 2 (6 + e2 )2a2 (1 e2 )2 P

+15 r 3 (54 6e + 15 e2 2e3 )

2a3 (1 e2 )3 P . (61)

What is the meaning of e in the solution wn of ordern ? In the zero-order solution w0 , e is the eccentricity,

but in the successive approximations it loses this charac-terization: the symbol e is simply a constant in the openinterval (0,1) that one introduces in the initial conditionsto express the shape of the orbit, and that coincides withthe eccentricity of the osculating Kepler ellipse to thepath followed by the planet when = 0.

III. APPLICATIONS

In astrophysical applications Eq. (61), written in theform

f ( r ,a,e, ) = 0 , (62)

is an implicit relation between dynamical and orbital pa-rameters characterizing the system under consideration,and it can be used to calculate any of them, once knownthe others. Thus, for example, in the solar system r , a ,eare known, and we calculate , the perihelion shift of the planetary orbits. It is also evident that a periastronadvance is highly enhanced by small as (whence short

periods) and high orbital eccentricities. A word of cau-tion is needed here: we must consider the fact that a verylarge eccentricity can also mean a very small periastrondistance a(1e), and so we must stop precisely at the or-bit that just grazes the surface of the star. The problemof detecting the motion of periastron (or of apoastron) of some highly elliptic extrasolar planets has been consid-ered by some authors 9,17 , and of course now the questionis to determine the magnitude of the higher orders effectsfor some plausible orbital parameters. In the rst columnof Table I for comparison purposes we have inserted thedata concerning the Sun-Mercury system, while the sec-ond and third columns are referred to two hypothetical

exoplanets of a solar-mass star with great eccentricitiesand/or small radial distances. While is doubtful, giventhe particularly high levels of observational accuracy re-quired, that it is actually possible to nd planets with or-bital characteristics tted for this purpose, in meantimeone can imagine a verication of the high-orders peri-helion formula achieved by means of a man-made solarprobe in a carefully planned celestial mechanics experi-ment. However, we believe that the higher order termsin Eq. (61) must be taken into account in all attemptsto detect the Suns Lense-Thirring effect on the periheliaof the inner planets in order to separate with improved


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TABLE I: Mercury and two hypothetical exoplanets

System Sun/Mercury Star/Alpha Star/Beta

M 1.00 1.00 1.00 (cm) 1 .475 105 1.475 105 1.475 105

a (cm) 5 .791 1012 5.791 1012 8.788 1010

e 0.2056 0.95 0.20

1.038 10 5

1.038 10 5

1.038 10 5

P (d) 87.9 87.9 0.1641 (rad/d) 5 .703 10 9 5.602 10 8 2.001 10 4

2 (rad/d) 1 .097 10 15 1.262 10 13 2.652 10 9

3 (rad/d) 2 .46 10 22 2.873 10 19 4.098 10 14

1 (arcsec/yr) 0 .429 4.220 1.514 104

2 (arcsec/yr) 2 .26 10 10 9.51 10 6 1.998 10 1

3 (arcsec/yr) 5 .09 10 17 2.16 10 11 3.088 10 6

accuracy these two important consequences of general

relativity.Some close binary systems are good subjects for a ver-

ication and an application of the formulas 12 . Beforeapplying Eqs. (60), (61) to a concrete example, we re-call that a generalization of Ensteins perihelion formulasays that in a system of two non negligible sphericalmasses, the two-body problem, the gravitational param-eter means GM = G(m1 + m 2 )13 . In this instancer could be loosely named the equivalent or the nominal

gravitational radius of the system. This result is onlythe rst-order approximation to a full relativistic treat-ment of the center of mass of two comparable bodies, ahitherto unresolved problem, but however we will fullyexploit the effect of this approximation. For these sys-tems, the semimajor axis and the total mass M are notdirectly determinable, while eccentricities, periods andperiastron advance rate are, so we can use Eq. (62) tond r , once we have replaced a with the period P andthe total mass M (the degeneracy of the mass is removedby measurement of another relativistic orbital character-istic, the Einstein parameter) using Keplers third law inthe form

a =P 2 GM

42

1 / 3

=P c

2

2 / 3r 1 / 3 . (63)

To be precise, this law is not exactly true in a regimeof strong elds and little distances, but it can be usedharmlessly in a perturbative calculation for the systemswe are considering within the same constraints we haveset for the center of mass, that is a rst-order approxi-mation. Incidentally, this will free our results from othercomplex and still poor understood effects and, besides,this is in line with the regime of validity of the equationof motion which we have supposed. From Eqs. (61), (63)

we get

=2P

5 / 3r

c

2 / 3

f (e) +2P

7 / 3r

c

4 / 3

g(e)

+2P

3r

c

2

h(e), (64)

where

f (e) =3

(1 e2 ),

g(e) =15(6 + e2 )2(1 e2 )2

,

h(e) =15(54 6e + 15 e2 2e3 )

4(1 e2 )3.

To obtain directly the value of the mass M in units of the solar mass M , we can replace in Eq. (64) the ratior /c with the product T M , where T GM /c 3 is themass of the Sun expressed in units of time. In the cur-

rent literature the expression of lacks of the last twoterm of Eq. (64). 12 Inserting the values of e,P, deter-mined for a given binary system, Eqs. (64), (61) can benumerically solved for r and M respectively, and willgive the value of the gravitational radius of the systemor the total mass. Once known r , we can complete thecalculation and use Eq. (61) to compute the value of a.The value of that one determines is due, in general,to relativity plus extra classical terms, as the gravita-tional quadrupole moment induced by rotation and tidaldeformations. But strongly self-gravitating objects as bi-nary pulsars have a mass pointlike behavior, and thus themotion of their periastron must be entirely ascribed torelativity. Strictly speaking, since this bodies are rapidlyspinning, one should consider also the Lense-Thirring ef-fect or use the Kerr solution to the Einstein equation,but this is an argument for a further study.

We apply the theory to the double pulsar J0737-3039.Its orbital period is the smallest so far known for suchan object, and it can be determined with great precisionalong with . For a such system cumulative effects addrapidly, and this allows a meaningful application of theformulas, despite the rather small eccentricity. In Table

TABLE II: Pulsar J0737-3039

e 0.0877775P 0.10225156248(d) 8834.534991(s) 16.89947o (yr 1 ) 9.346445651 10 9 (s 1 )

c 2.99792458 1010 (cm/s)T 4.925490947 10 6 (s)yr 3.15576 107 (s)

II we have indicated the determined 12 values of e,P, and, for the readers convenience, some numerical valuesused in the calculations, while in Table III are indicatedall parameters that can be derived through an application


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TABLE III: Pulsar J0737-3039 derived parameters

To order: 3

r (cm) 3 .82014 105 3.8199525 105

M (M ) 2.587075 2 .586948a (cm) 8 .788391 1010 8.788680 1010

1.314166 10 5 1.314057 10 5

1 16.89947o

yr 1

16.89891408o

yr 1

2 - 0.00055589o yr 1

3 - 0.00000002o yr 1

16.89947o yr 1 16.89946999o yr 1

of our formulas. In the rst column, the computation aredone to order , assuming the validity of general relativ-ity thorough the use of Einsteins precession formula 1of Eq. (61), while in the second column are inserted thevalues we have computed to order 3 , in particular howmuch of comes from i , i = 1 , 2, 3. Since from the ob-servational point of view the single higher-orders contri-butions to the precession rate are inextricably combined,and so hidden to a direct measurement, we can obtainan indirect verication turning to the main consequenceof this approximation: a diminution by a small amountof the total mass of the system with respect to the cur-rently accepted value, with the consequent redenition of other parameters, in particular of a, and so we can re-ne the model of the system with one that ts the foundvariations.

Ever more accurate determinations of might enable,for this as for any other similar system, to test the use-fulness of the third-order approximation to the perias-tron secular motion deduced from the equation of mo-tion within the limits of approximation that we have in-troduced. However, the results found can be consideredstrictly true for the motion of a test body in the gravi-tational eld of a central body of mass equivalent to thetotal mass of the binary system.

IV. APPENDIX

By simple considerations of powers and arguments,and with the aid of the subscript notation, we can quicklynd the coefficient of the secular term H n once knownH 2 , . . . , H n 1 and the orbit to order n 1 . Here is asketch of how we unearth from the right side of Eq. (54)

(1 + w22 ) H 2e

w2 = (1 + w22 ) (2A1 + B 1 )w2 , (65)

the resonant term H 3 cos k 2 of order 3 .We consider rst the multinomial expression

w22 = [A1 + A2 + ( e + E 1 + E 2 )cos k 2 + ( B1 + B 2 )cos2k 2 + C 2 cos3k 2 ]2 ,

(66)

together with the following algebraic and trigonometricalidentities

(a + b + c + . . . )2 = a2 + b2 + c2 + . . .

+ 2 ab+ 2 ac+ . . . + 2 bc+ . . . , (67)

(cos nk 2 )2 =12

cos2nk 2 +12

, (68)

a cos(n + 1) k 2 bcos nk 2 =ab2

cos k 2 + . . . . (69)

Heres the argument: in expanding Eq. (66) accordingto formula (67) we can omit to explicitly writing thecosines, since we know that each e, E multiplies cos k 2 ,so as each B and C multiplies cos 2 k 2 and cos 3k 2 re-spectively. This way we can proceed rapidly by inspect-ing more concise expressions. Let us lter now Eq. (66)through the sieve represented by the constraints we haveimposed to isolate the resonant term

3 cos k 2 .

We observe rst that the squared terms in Eq. (67)can be deleted because of Eq. (68) for n = 1 , 2, 3. Next,we drop all double products of Eq. (67) in which thesum of the subscript indices is different from 2 and so,after multiplication by , will survive only the terms of order 3 . Last, when we meet products of two cosines,we consider only those in which the arguments differ byone, and apply to them Eq. (69). At the end we obtainthe following sum of coefficients of cosk 2

2 eA2 + eB2 + 2 A1 E 1 + E 1 B1 , (70)

but the last two terms are erased by the 3 -coefficientsarising from the rightmost expression of Eq. (65), whichare the constants 2 A1 and B1 times the termE 1 cos k 2 of w2 , and thus we nally get

H 3 = e(2A2 + B2 ). (71)

Electronic address: [email protected] Bertotti B., Farinella P., Vokrouhlicky D., 2003. Physics

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